The Put–Call Parity Identity

Why Put–Call Parity Holds

A European call and a European put with the same exercise price and expiration respond differently to the stock price at expiration. Yet each can be combined with another asset to produce the same total payoff. That equality gives a precise relationship among the options’ current values, the stock price, and the present value of the exercise price.

The central point is that the exercise price is paid or received at expiration, not today. Its current value is therefore its discounted value. Matching the payoffs first makes clear both why the relationship holds and where that discounting belongs.

Definition: Let $S_0$ be the stock price today, $S_T$ the stock price at expiration $T$, $C_0$ the current value of a European call, and $P_0$ the current value of a European put. Suppose both options are on one share of the same non-dividend-paying stock, have the same exercise price $K$ and expiration $T$, and the effective annual risk-free rate $i$ applies over the period. Put–call parity is $$ C_0+K v^T=P_0+S_0,\qquad v=\frac{1}{1+i}. $$

The left side is the current cost of a call plus an investment that will pay $K$ at expiration. The right side is the current cost of a put plus one share of stock. The identity applies to European options because their exercise is restricted to expiration. It also assumes that the options have matching terms and that the stock pays no dividends before expiration.

Match the Expiration Payoffs

At expiration, a European call with exercise price $K$ pays $\max(S_T-K,0)$. A European put with the same exercise price pays $\max(K-S_T,0)$. The cash investment in the call-plus-cash portfolio is chosen to pay $K$ at expiration; its cost today is $K v^T$. The put-plus-stock portfolio contains one share, worth $S_T$ at expiration.

To compare the portfolios, consider whether the stock price ends above or below $K$. If $S_T$ is at least $K$, the call is worth $S_T-K$, while the put is worth zero. If $S_T$ is below $K$, the call is worth zero, while the put is worth $K-S_T$. In either case, adding the cash payoff to the call or the stock value to the put gives the same total.

\max(S_T-K,0)+K=\max(K-S_T,0)+S_T

This equality of expiration payoffs justifies the price identity. If two portfolios have identical payoffs in every possible state at expiration, they must have the same value today in a market with no arbitrage. Otherwise, one could buy the less expensive portfolio and sell the more expensive one, while the matching future payoffs would offset. Since the cash payment $K$ occurs at $T$, the cost of that cash investment today is its present value $K v^T$, not $K$.

Write each terminal payoff.
The call-plus-cash payoff is $\max(S_T-K,0)+K$; the put-plus-stock payoff is $\max(K-S_T,0)+S_T$.

Check both possible price regions.
For $S_T\ge K$, each payoff is $S_T$. For $S_T<K$, each payoff is $K$.

Equate current values.
Equal payoffs imply equal current values under no arbitrage: $C_0+K v^T=P_0+S_0$.

Worked Payoff Comparison

A payoff table is a direct way to see the replication. It also prevents a frequent error: comparing the call and put alone while forgetting the cash or stock included in each portfolio.

Worked Example: Comparing Expiration Payoffs in Three Stock-Price States

A non-dividend-paying stock has European options with $K=\$100$ and expiration at time $T$. Compare a call plus an investment that pays $\$100$ at expiration with a put plus one share of stock, for $S_T=\$70$, $\$100$, and $\$130$.

Method. Calculate each option’s expiration payoff, then include the additional asset in its portfolio. No interest-rate calculation is needed to compare expiration payoffs.

If $S_T=\$70$. The call pays $\max(70-100,0)=\$0$, so the call-plus-cash portfolio pays $\$0+\$100=\$100$. The put pays $\max(100-70,0)=\$30$, so the put-plus-stock portfolio pays $\$30+\$70=\$100$.

If $S_T=\$100$. The call and put each pay $\$0$. The call-plus-cash portfolio pays $\$0+\$100=\$100$, and the put-plus-stock portfolio pays $\$0+\$100=\$100$.

If $S_T=\$130$. The call pays $\max(130-100,0)=\$30$, so its portfolio pays $\$30+\$100=\$130$. The put pays $\max(100-130,0)=\$0$, so its portfolio pays $\$0+\$130=\$130$.

Check. The two portfolios pay the same amount in all three states: $\$100$, $\$100$, and $\$130$, respectively. The method avoids the mistake of comparing option payoffs without including the portfolio’s cash or stock.

From Payoff Equality to Current Values

Payoff matching gives the equation’s structure; discounting gives the correct amount to put in the equation. Under an effective annual rate $i$, one dollar invested today grows to $1+i$ dollars after one year. Thus the amount invested today that will grow to $K$ after $T$ years is $K/(1+i)^T=K v^T$. This uses the effective annual rate over a period of $T$ years; multiplying the annual rate by $T$ is not the same calculation.

\begin{aligned} \text{Current value of call plus cash} &= C_0+K v^T,\\ \text{Current value of put plus stock} &= P_0+S_0. \end{aligned}

Because the portfolios have the same expiration payoff in every state, the no-arbitrage principle equates their current values. That produces put–call parity. No assumption about the probability distribution of $S_T$ is required: the payoff equality holds for every possible expiration stock price.

Worked Example: Verifying Parity from Current Portfolio Values

A non-dividend-paying stock is priced at $\$105$. A European call and put both have $K=\$110$ and $T=1$ year. The effective annual risk-free rate is $i=10\%$. Suppose the call is worth $\$18$ and the put is worth $\$13$. Verify that the two portfolios have the same current value.

Method. Discount the exercise price for one year, then add the current value of each portfolio’s option and other asset.

The discount factor is $v=1/(1+0.10)=1/1.10$. Therefore, the present value of the $\$110$ expiration payment is $$ K v^T=\frac{110}{1.10^1}=\$100. $$

The call-plus-cash portfolio is currently worth $$ C_0+K v^T=18+100=\$118. $$

The put-plus-stock portfolio is currently worth $$ P_0+S_0=13+105=\$118. $$

Check. Both portfolio values equal $\$118$, as parity requires. The method avoids treating the $\$110$ due in one year as though it were worth $\$110$ today.

Using an Effective Annual Rate Over More Than One Year

When expiration is more than one year away, the present value calculation must reflect the full term. For example, with an effective annual rate, a cash amount due in two years is discounted by $(1+i)^2$, not by $1+2i$. Keep full precision during the calculation and round only the final reported dollar amount.

Worked Example: Discounting the Exercise Price for Two Years

A non-dividend-paying stock is priced at $S_0=\$100$. A European call and put have $K=\$100$ and $T=2$ years. The effective annual risk-free rate is $i=5\%$, and the call is worth $\$10$. Use parity to find the put value implied by these inputs, then verify both portfolio values.

Method. First calculate the two-year present value of the exercise price. Rearrange parity to express the put value, and substitute the result back into both sides as a check.

The two-year accumulation factor is $$ (1+i)^T=(1.05)^2=1.1025. $$

The present value of the exercise price is $$ K v^T=\frac{100}{1.1025}=\$90.7029478458\ldots. $$

Rearranging $C_0+K v^T=P_0+S_0$ gives $$ P_0=C_0+K v^T-S_0 =10+90.7029478458\ldots-100 =\$0.7029478458\ldots. $$

The call-plus-cash portfolio is worth $10+90.7029478458\ldots=\$100.7029478458\ldots$. The put-plus-stock portfolio is worth $0.7029478458\ldots+100=\$100.7029478458\ldots$. Rounded to the nearest cent only at the final step, the implied put value is $\$0.70$ and each portfolio is worth $\$100.70$.

Check. Using the unrounded put value, both sides are equal. Rounding the put before checking may create a small apparent difference; retaining precision until the final step avoids that issue.

Assumptions and Common Exam Traps

Put–call parity is a relationship for matching European options on a non-dividend-paying stock. Before applying it, confirm that the call and put have the same underlying share, exercise price, and expiration. The identity also relies on the ability to trade the stock, options, and risk-free cash consistently, without transaction costs or restrictions that prevent the relevant positions. The risk-free rate must be appropriate for the time to expiration.

Using $K$ instead of $K v^T$. The exercise price is paid at expiration, so it must be discounted to compare its value today with the current stock and option prices. For example, in the one-year example above, incorrectly using $K=\$110$ gives a call-plus-cash value of $\$18+\$110=\$128$, rather than the correctly discounted $\$118$.
Using the wrong time unit. With an effective annual rate, the discount factor for $T$ years is $v^T$. For two years at $5\%$, use $(1.05)^2=1.1025$, not $1+2(0.05)=1.10$.
Leaving out one asset. The parity portfolios are call plus cash and put plus stock. The options alone do not have equal payoffs in general.
Applying the identity to mismatched contracts. Different exercise prices or expiration dates do not produce the payoff equality used in the derivation.
Ignoring dividends or early exercise. A dividend-paying stock changes the cash-flow comparison, and American options can be exercised before expiration. The identity stated here is for European options on a non-dividend-paying stock; it should not be transferred unchanged to those other settings.

The identity is useful because it links option values to assets and cash flows whose roles are explicit. For exam calculations, the safest sequence is to check the assumptions, match the contracts, discount $K$ over the stated term, and then compare the two complete portfolio values. This keeps the payoff logic separate from the arithmetic and makes errors in timing or portfolio construction easier to spot.

Check Your Understanding

For each problem, use the matching-contract and non-dividend assumptions stated in the question. Round dollar answers to the nearest cent when needed.

A non-dividend-paying stock has European options with $K=\$75$. At expiration, calculate the call-plus-$\$75$-cash and put-plus-stock payoffs when $S_T=\$60$, $\$75$, and $\$95$. What are the three matching payoff amounts? (Answer: $\$75$, $\$75$, and $\$95$, respectively.)
A European call and put have the same $K=\$200$ and expire in two years. The effective annual risk-free rate is $5\%$. Calculate the present value today of the cash payment due at expiration. (Answer: $\$181.41$.)
A non-dividend-paying stock is worth $\$62$. A European call with $K=\$60$ and $T=1$ year is worth $\$8.50$. If the effective annual risk-free rate is $5\%$, what put value is implied by parity for a put with the same terms? (Answer: $\$3.64$.)
A European call and put on a non-dividend-paying stock have $K=\$250$ and expire in half a year. The effective annual risk-free rate is $10\%$. Calculate the present value of the exercise price, using the effective annual rate over $T=0.5$ years. (Answer: $\$238.37$.)
A non-dividend-paying stock is worth $\$140$. A European call with $K=\$150$, $T=1$ year, and value $\$12$ is paired with a European put on the same terms. At an effective annual risk-free rate of $4\%$, calculate the put value implied by parity. (Answer: $\$16.23$.)

What You'll Learn